Qualitative Fuzzy Sets and Granularity

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1 Qualitative Fuzzy Sets and Granularity T. Y. Lin Department of Mathematics and Computer Science San Jose State University, San Jose, California and Shusaku Tsumoto Department of Medical Informatics Shimane Medical University, 89-1 Enya-Cho Izumo City, Shimane , Japan Abstract This paper proposes qualitative fuzzy sets, where each membership function is able to tolerate small amounts of perturbations. This can be viewed as one of the type II fuzzy sets, where the grade of a membership function is represented by a fuzzy number. In this paper, we surveyed our qualitative fuzzy sets from the viewpoint of category theory. 2 Categories - Universes of Discourse Fuzzy theories have been driven by applications, So each theory has its unique territory or universe. Mathematicians use the notion of categories to formally define their universe. We will use such a language to describe various universes. Roughly a category consist of 1. A class of objects, and 1 Introduction Intuitively, a fuzzy membership should be able to tolerate small amounts of perturbations on the degree of its membership. However, a traditional fuzzy set is defined by a unique membership function([18], pp.12). So, each perturbation of the given membership function is a new function and hence defines, based on the traditional definition, a new traditional fuzzy set (Figure 1.) This obviously cannot be the original intuitive notion of fuzzy sets. In fact, L. Zadeh proposed, at the very beginning, type II fuzzy sets, whose memberships are expressed by fuzzy numbers. Each fuzzy number is a granule (a traditional fuzzy set) attached to every real number. Recently, there are renewed interests on this topic. Nevertheless, as our main interests are in the foundational issues, we will restrict to the discussions of the original ideas. We have proposed several geometric versions, which intend to accommodate the physical perturbation of membership functions; Each version has various degrees of success and failure. Proc. 7th IEEE Int. Conf. on Cognitive Informatics (ICCI'08) Y. Wang, D. Zhang, J.-C. Latombe, and W. Kinsner (Eds.) /08/$ IEEE For every ordered pair of objects X and Y, a set Mor(X, Y ) of morphisms, which satisfies certain properties. The details are not important for our purpose. We merely need a concenient language. 2.1 Category of Cantor Sets The objects are Cantor sets. A morphim is the set of all (discrete) maps between Cantor sets. 2.2 Category D 0 of TNS (Topological Spaces) The objects are the topological spaces, and the morphisms are continuous maps between spaces. We will assume all topological are rather nice, namely; they are all locally compact Hausdorff spaces. Intuitively a topological space is a Cantor set in which nearness makes sense. Real numbers, Euclidean spaces are topological spaces. Formally, a topological space is a Cantor set in which each point p is assigned a family of subsets, which satisfies some axioms. Each such a subset is called a (topological)

2 Figure 1. Perturbated Membership Function. Conventional View: a perturbated membership function is different from original one. Qualitative fuzzy sets view these as the same. neighborhood N P of p. The family of all such topological neighborhoods at p (NS(p)) is called a topological neighborhood system at p. The totality of all the neighborhood system at every p is called a topological neighborhood system(tns(u)) on the universe U. Please note TNS space is equivalent to the usual topological space That is defined by open sets. 2.3 Category D k of Differentiable Spaces This is a sub-category of topological spaces. All topological spaces and maps should be differentiable up to the order k, k =0, 1,..., ; D 0 is the topological category. Most of the universes of fuzzy control live in this class of categories. 2.4 Category of Measurable Spaces This is a category, where length, area, volume or its generalizations make sense. We did develop a qualitative fuzzy theory in this category [5]; the universes of fuzzy control live in here too. 2.5 Category of LNS(Granular Sets) In many applications of topological space( e.g. numerical analysis), one does not need the full topology, often an instance of topology(e.g., given a fixed ɛ neighborhood, or radius of error). Based on this observation, we have proposed LNS-spaces, each of which consists of a set together with LNS(Less than topological Neighborhood Systems). Now we shall consider the category of LNS-spaces. The objects are LNS-spaces or granular sets, and a morphim is a set of continuous maps. This is a generalization of topological spaces. The only differences between LNS and TNS is that LNS does not impose axioms on NS(p), at p. NS(p) maybeemptyandn p may not contain p. This category is implicitly used in numerical analysis. In fact, we do consider more refined notions, all neighborhood spaces are indexed by a fixed space, call object space and LNS are called the data spaces. Definition 1 Let V and U be two universes (classical sets). For each object p V, we associate a family of crisp/fuzzy subsets, denoted by NS(p) P (U), that is, we have a map B : V 2 P (U) : p NS(p), where P (U) means all crisp/fuzzy subsets on U, and 2 P (U) means the power set of all crisp/fuzzy subsets. NS(p) is called a crisp/fuzzy neighborhood system at p. NS(U)={NS(p) p V } is a neighborhood system of V. Also we have considered the fuzzified spaces. 2.6 Category of BNS (binary relations). This is a special case of previous category. A neighborhood system in which there is at most one neighborhood (may be empty) at each point p is called a binary neighborhood system. A binary neighborhood system defines a binary relation and vice versa [7]. In the case NS(p)={B(p)} is a crisp/fuzzy singleton, we define 436

3 Definition 2 The map B: V P(U), binary granulation (BG) and the collection {B(p) p V } a crisp/fuzzy binary neighborhood system (BNS). We should observe that B(p) is a neighborhood (granule) of p implies p B(p), but it does not imply that q B(p) implies B(p) is a neighborhood of q. It is clear that given a map B gives rise to a crisp/fuzzy binary relation BR: BR = {(p, x) x B(p) and p V }. In both crisp/fuzzy cases, BR, BNS, and BG will be treated as synonyms. 2.7 Categories of Fuzzy Sets The unit interval [0, 1] plays very important role in fuzzy theory, so we present a precise version of it for every category mentioned above. In the category of sets, The unit interval [0, 1] is a discrete set of numbers lying between 0 and 1. In the topological category, the unit interval has the so called usual topology; it is what we normally used in the college mathematics. In the category of LNS, we choose TNS. In the category of BNS, we choose a fixed ɛ and consider an ɛ-neighborhood at every point. In the category of measurable space we use Lebesque measure on [0,1]. Every category has one fuzzified version; they are denoted by FLNS, FBNS and etc. 2.8 Some Basics Lemma 1 Let B A be the set of maps between A and B. Then we have the following exponential law: (C B ) A C A B Instead of giving a formal proof, we will illustrate the idea in examples. Let P, Q be sets. Consider z : P [0, 1] Q ; p f, where f : q t [0, 1], that is, t = f(q). Themapz induces the following map z :(p, q) t = f(q), To see the converse, at z, we keep the p fixed and let q varies, then we have a map q t [0, 1]; let it be f. Now observing that we just, for each p, find an f. That is, there is a map from P to functions in [0, 1] Q. This lemma are well known in the set category. From ([15], pp. 6), we have the following topological version: Lemma 2 If A is a locally Hausdorff space, C is a Hausdorff space and B is a topological spaces, a map g : C B A is continuous if and only if the composition, C A B A A B is continuous, where the last map is evaluation map. 3 Soft Sets - Traditional Fuzzy Sets We need to use traditional fuzzy sets to describe qualitative fuzzy sets. So the term fuzzy set may take several meanings; to avoid confusing, we will observe the following convention: 1 First, the term fuzzy set will have no technical content, it will refer to the intuitive vague concept; its precise meaning is the goal of this paper. We have given the traditional fuzzy set a new term W - sofset in [10]; we will use the new term. Formally, 2 A W-sofset is the mathematical object defined uniquely by a membership function. We will simply refer to them as softsets; previously we used sofsets, where we deliberate removed the t. 3 Traditional fuzzy numbers are softsets on real numbers that meet some criteria that are specified in [18]. However, we need some kind of continuity or topology among nearby fuzzy numbers; so we will define the unit interval of softnumbers 4 The unit interval of softnumber is the image of the map FS :[0, 1] Mor([0, 1], [0, 1]), that map each real number to its softnumber that is a member in Mor([0,1], [0,1]), where Mor([0,1], [0,1]) is equipped with compact open topology. By Lemma 2, Section 2.8, FS is equivalent to the following continuous map (and vice versa) 5 z : U I [0, 1]. Definition 3 In the differentiable categories, we say FS is differentiable, if z : U I [0, 1] is differentiable. 6 A slice S t = FS t (t, ) =FS(t, ) is a softnumber in the unit interval of sofnumbers. To avoid ambiguity, we will call S t the softnumber, representing t [0, 1]. 4 Qualitative Fuzzy Sets In control, the universe of discourse U is more than a discrete set, it needs the notion of continuity and differentiability. Traditionally, topological spaces is the right space to talk about continuity. However, in computer science, topological space structure is over kill. LNS, which is a generalization of topological spaces, has been proposed to support the intuitive idea of continuity. Since 1988 [12], [11]. From here on all spaces, including U, are LNS as explain in Section 2. Let MF(U) be LNS of all continuous membership functions. For simplicity we write V = MF(U). 437

4 4.1 Granular Computing Version In [10], one of us introduced a family of qualitative fuzzy sets, in which every qualitative fuzzy set is defined by a neighborhood N f in the membership function space MF(U). In that paper, LNS was defined on a single universe. In [7], [8],[6], a more elaborate theory of LNS, in which a neighborhood system was defined on two universes V, called object space, and U, called data space, is defined. We can view the two universe version as follows: each neighborhood in U is labeled (or indexed) by some object(s) in V : To each object p V (object space), we associate an (empty, finite or infinite) family of crisp/fuzzy subsets of U(data space). These subsets are called crisp/fuzzy basic granule of p, denoted by (LNS V (p)/f LNS V (p)). Note that if there is at most one (could be none) basic granule at every point p V, then LNS is BNS, which is definable by a binary relation. Now by considering MF(U),the membership function space, as data space, and following the spirit of [10], we define Definition 4 Each basic granule on MF(U), in notation t : V MF(U) =I U, defines a qualitative fuzzy set on U (indexed by object space V, called context by Thiele). Note that, by Lemma 1, this is equivalent to t : VXU I, In other words, t is a fuzzy binary neighborhood on U indexed by V. Figure 2 illustrates the construction of this binary neighborhood. 4.2 Thiele s Version According to Thiele, a context dependent fuzzy set is a map where W is the context. given in Section 4.1. t : WXU [0, 1], This is precisely the definition Proposition 1 The notion of Thiele s qualitative fuzzy set is equivalent to granular computing version of qualitative fuzzy set. This says granular computing approach is the same as the qualitative fuzzy set of Thiele [17]. 5 Type II Fuzzy Sets 5.1 Granular Approach We will write I =[0, 1] and use both in the case we want to have different copies. A type II fuzzy set is a map z : U [0, 1] I ; By Lemma, it is equivalent to a map z : U I [0, 1]. For each fixed r in I, this map z induces slices (of U), z r = z (,r):u [0, 1],z r(u) =t(= z (u, r)), Each slice is a softset. From Section 4.1, z induces a fuzzy binary neighborhood systems on U indexed by I and vice versa. Hence z is a qualitative fuzzy set in the sense of granular computing version. So the collection of the slice softsets induces type II fuzzy set z is a qualitative fuzzy set in the granular approach, namely, Proposition 2 A type II fuzzy set is a qualitative fuzzy sets on U indexed by I. This results led us to consider homotopy type qualitative fuzzy sets. 6 Stochastic Homotopy Version In a more geometric approach, we use deformations of membership functions to develop some qualitative theory [11], [9]. We have used homeomorphism, to do the deformations. Thiele pointed out weak points of our first approach [16]; we did know its weakness ourselves much earlier. In our final formulation, we used the homotopy [5]. This version gives us a feeling of physical deformation and perturbation; This approach may give us a more satisfactory theory; further works are needed. We also need measure theory on LNS-space. Let B be a Borel sets on the LNS-space U, and m a measure, then (X, B, m) is called a measure space [1]. We say X has property Y epsilon-nearly everywhere, denoted by n.e. if and only if X has property Y at every point but a subset of illegal points, whose measure is less than given epsilon. We say X has property Y almost everywhere, denoted by a.e. if ɛ is zero. We say X has property Y (absolutely everywhere) if illegal points is empty. We say a membership function FY V is ɛ-deformable from FX iff there exists a map (this is a homotopy) such that Φ:U I [0, 1] 438

5 FBNS(p): Fuzzy Binary Neighborhood System FB(p) FB(p): Neighborhood System Qualitative Fuzzy Set FB-Sofset. p V=MF(U): a set of membership functions x U: classical sets (Data, Objects) Figure 2. Fuzzy Binary Neighborhood System 1. Φ(u, 0) = FX(u) n.e. [or a.e.] 2. Φ(u, 1) = FY(u) n.e. [or a.e.] This defines a equivalence relation on V. Moreover, there is natural algebraic structure on the basic granule derived from the algebraic structure, such as bounded sum and multiplication of the unit interval ([15], pp. 24 and 34). There are many ways one can form the algebraic operations; we will not digress there, since no obvious applications in sight. Proposition 3 Homotopy forms a partition on MF(U), so the qualitative fuzzy sets in geometric approach is an P - qualitative fuzzy set; see [4]. 7 Conclusions Kandle [2] (pp. 4) used several membership functions to define a fuzzy set; His approach was intuitive. Since 1992 [11] of Lin s work, we have continued to work on the notion of qualitative fuzzy set. This paper give a slightly stronger links to various theories. References [1] Halmos, P., Measure Theory, Van Nostrand, [2] Kandel, A., Fuzzy Mathematical Techniques with Applications, Addision-Wesley, Reading Massachusetts, 1986 [3] Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, [4] Tsau Young Lin, Qualitative Fuzzy Sets: A comparison of three approaches. In: Proceeding of Joint 9th IFSA World Congress and 20th NAFIPS International Conference, Vancouver, Canada, July 25-28, 2001, pp ; [5] T. Y. Lin, and S. Tsumoto, Qualitative Fuzzy Sets Revisited: Granulation on the Space of Membership Functions, The 19th International Meeting of the North American Fuzzy Information Processing Society, July 1-15, 2000, Atlanta, pp [6] T. Y. Lin, Granular Computing: Fuzzy Logic and Rough Sets. In: Computing with words in information/intelligent systems, L.A. Zadeh and J. Kacprzyk (eds), Springer-Verlag, ,

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